First of all a good breeder doesn't add to any problems.
They will take back any dog they have bred at any point in the dogs life so it never ends up in rescue, they will be doing the necessary genetic/screening tests to guarantee freedom from inherited disease and they will be taking care to select parent dogs of stable friendly temperament and socialise the puppies which puts the chances of them staying in a happy home higher.
Your personal dogs might not have brought disease but a great many imported dogs have and do.
A number of vets have spoken out on the issue.
On the rescue spaces, what about all the successful overseas adoptions from people who were declined with kids who would have gone on to adopt a British dog at a later date and now won't?
Or at the other end of the spectrum the people who adopted a gone wrong foreign and are now sworn off rescue completely?
In countries like Spain, Cyprus etc dogs are not valued, failed hunting dogs, unwanted pets etc are cast out entire on the street and keep breeding and breeding.
You cannot physically take enough of these dogs abroad to stop the problem of dogs on the street.
You are talking about millions and millions of dogs.
Many of which are badly traumatised and will never make good pets.
Soon as a group of dogs are shipped more are abandoned or birthed on the streets.
In order to stop the problem the street dogs need to be neutered and heavy penalties need to be put in place to punish those who dump and/or abuse them and people need educating as to why their behaviour is wrong.
Until that happens dogs will continue to be abandoned and abused.
Shipping them abroad is nice for those few dogs (who don't end up PTS) but it does nothing at all to tackle the actual problem.
I suppose there is a case for the immediate cessation of breeding until the rescues have found homes but then Britain has a different dog problem to the overseas countries.
Britain doesn't have a culture of street dogs out breeding in need of neutering but it does have a large population of people that breed substandard dogs for all the wrong reasons.
The answer imo would be tighter regulation on breeders.